A complete mappingof a finite group $G$ is a permutation $\phi: G\rightarrow G$ such that$x\mapsto x\phi(x)$ is also a permutation. A permutation $\theta:G\rightarrowG$ of a finite group $G$ is an orthomorphism of $G$ if the mapping $x\mapstox^{-1}\theta(x)$ is also a permutation. Two orthomorphisms $\theta$ and $\phi$of $G$ are orthogonal if the mapping $x\mapsto \theta(x)^{-1}\phi(x)$ isbijective. This talk provides a brief introduction to the existence of completemappings and orthogonal orthomorphisms of groups, focusing on their algebraicand extremal aspects. Difference matrices have a close relationship with orthogonalorthomorphisms of groups. It will also give a survey on difference matrices andtheir related topics, such as orthogonal arrays and mutually orthogonal Latinsquares.

Tao Feng receivedthe B.S. and Ph.D. degrees in mathematics from Beijing Jiaotong University,China, in 2003 and 2008, respectively. He is currently a Professor with theSchool of Mathematics and Statistics, Beijing Jiaotong University. His researchinterests include combinatorial design theory and coding theory.